Neutrino mixing and masses in SO(10) GUTs with hidden sector and flavor symmetries

We consider the neutrino masses and mixing in the framework of SO(10) GUTs with hidden sector consisting of fermionic and bosonic SO(10) singlets and flavor symmetries. The framework allows to disentangle the CKM physics responsible for the CKM mixing and different mass hierarchies of quarks and leptons and the neutrino new physics which produces smallness of neutrino masses and large lepton mixing. The framework leads naturally to the relation $U_{PMNS} \sim V_{CKM}^{\dagger} U_0$, where structure of $U_0$ is determined by the flavor symmetry. The key feature of the framework is that apart from the Dirac mass matrices $m_D$, the portal mass matrix $M_D$ and the mass matrix of singlets $M_S$ are also involved in generation of the lepton mixing. This opens up new possibilities to realize the flavor symmetries and explain the data. Using $A_4 \times Z_4$ as the flavor group, we systematically explore the flavor structures which can be obtained in this framework depending on field content and symmetry assignments. We formulate additional conditions which lead to $U_0 \sim U_{TBM}$ or $U_{BM}$. They include (i) equality (in general, proportionality) of the singlet flavons couplings, (ii) equality of their VEVs; (iii) correlation between VEVs of singlets and triplet, (iv) certain VEV alignment of flavon triplet(s). These features can follow from additional symmetries or be remnants of further unification. Phenomenologically viable schemes with minimal flavon content and minimal number of couplings are constructed